(1) Human Nature: Rand, like Sartre, believed in a radical conception of human freedom: human beings not only were free in the sense that they could make choices between competing alternatives, they were also free about their own psychological make-up. Human beings, upon birth, were both cognitive and emotional blank slates. Man was a being "of self-made soul," as she liked to put it. This extreme version of the free will doctrine is not supported by either the testimony of great historians and poets nor does it pass muster with the sciences of human nature (psychology, neuroscience, biology, etc.). It simply does not correspond to reality. Human beings are in fact born with certain innate tendencies of character. And not only that, their fundamental character and personality is affected, not merely by genetics, but even by such non-volitional phenomena such as lesions in the brain. Rand's theory of human nature simply cannot stand up to the facts of the case. It is a species of radicalism, not so very different from the social constructivism of the Left. Though not socially constructivist, it does embrace a kind of egocentric or mind-centric constructivism.
(2) History. Rand contended that history was determined by the broad abstractions of philosophy, particularly epistemological and "metaphysical" abstractions. She believed, for example, that Kant's Critique of Pure Reason helped bring about the holocaust, and that her philosophy would lead to renaissance of reason and capitalism. This theory of history as one of Rand's worst blunders. You have to be very ignorant of historical sociology to take such a theory seriously. I suspect it is little more than a rationalization of her own immense conceit. She wanted to believe that she could change the course of history. Hence the theory
(3) Ontology. Rand presented herself as a realist (i.e., she believed, as we all do, that an external world exists independent of our perception of it), but she rejected the dualistic implications of realism, as exemplified by Arthur Lovejoy in his seminal The Revolt Against Dualism. Although acknowledging that realism cannot be proved, she did believe it could be validated through the use of several axioms. The implication in some Objectivist writings, especially of a strictly Randian, orthodox cast, is that many important facts about reality can be deduced from these axioms, most of which are merely empty tautologies, such as "existence exists" and "A is A." In other words, despite her formal realism, at its core, Rand's philosophy is intransigently rationalistic. She attempts to determine matters of fact through logical, moral, and rhetorical constructions. Yet she insisted all along that all valid human ideas can be traced to some perception or observation! Obviously, logic was not her strong point.
(4) Religion. Rand was a militant atheist. She regarded religion as irrational and immoral. She believed that human beings could know for certain that God doesn't exist and that anyone who disagreed was not being rational or honest. When you read Objectivist writings on religion, which are extremely insensitive and utterly clueless, it is clear that neither Rand nor any of her followers understood a thing about religion. They viewed the phenomenon entirely from the outside, as they had found it in the summaries of theologies and in the pronouncements of religious extremists. They had no experience of what religion is for practicing believers. That entire world they kept themselves shut out of, as if afraid of contamination. But without firsthand experience of how religion works in the hearts and minds of devout religious believers, you can never understand religion. I say this irrespective of where you stand on the question of God's existence.
(5) Human Cognition. An important subject for Rand. She took great pains to develop a theory of concepts which (or so she claimed) solved the so-called "problem of universals." Scott Ryan, in his book Objectivism and The Corruption of Rationality, has shown that Rand didn't even understand the problem, let alone solve it. But that's not the least of what is wrong with Rand's theories of cognitionher epistemology, if you will. Rand believed that the basic facts of human cognition could be laid bare through careful introspection of how the mind works. Here she committed a major blunder. Cognitive science has discovered that most of what happens in the mind occurs well below the threshold of consciousness. If you want to know how the mind works, you have to observe and study other minds, which cognitive scientists have been doing now for the last three decades. The results of these studies conflict in many important ways with the assertions of Rand. For example, Rand stressed logic as an instrumental part of thinking. Cognitive scientists have found, however, that logic is not much used in real world thinking. Most thinking involves making educated guesses based on one's experience, and then testing these guesses empirically. That, at least, is how critically inclined professionals such as doctors and lawyers think. Rand's epistemology is no longer relevant. It lacks empirical backing. It based on little more than rationalistic speculation.
(6) Morality. Rand famously supported selfishness in her ethical writings. But she also attempted to frame a "rational" morality, that is, one based on "reason." Her theory is not very impressive. She begins by arguing for a survivalist morality. The good is what helps people to survive, the bad is what brings about death. But Rand did not really want to advocate a survivalist morality, because she could imagine situations in which survival would not be worth it (e.g., survival under terrible pain). So she introduced a qualification to her theory: not survival per se, but the "survival appropriate to a rational being." Well that changes everything! But what does it mean? How does one distinguish a survival appropriate to a rational being from one that isn't? Here is where the theory fails: Rand has no answer to this question. Her standard is so vague that it simply opens the door to casuistry.
(7) Aesthetics. I wrote in my book Ayn Rand Contra Human Nature that Rand aesthetics is little more than a rationalization of her phillistinism. The fact is, Rand really didn't like art. She despised most of the great artists of Western Civilization, including Shakespeare, Rembrandt, Tolstoy, and Van Gogh. She rationalized her dislikes by calling any art she didn't approve of as "malevolent" or deterministic.
Philosophically, Rand was a mess. Although undeniably brilliant and a gifted and charismatic writer, Rand was simply too egotistic to accept either the criticism of others or the evidence of scholars and scientists. She had to create a philosophy on her own resources alone, and other than IQ, she didn't have much to offer. She was not very well read nor did she have much experience with the greater world. She was an intellectual in the worst sense of the word: insular, provincial, uncurious, interested merely in projecting her own wishes and pathologies on thousands of adoring fans. It is not surprising that her star is beginning to slip, and that fewer and fewer people are falling under her spell. Although ome well meaning followers are trying to purge her philosophy of its worse elements by mixing it with a heady dose of common sense, this will not likely help Rand's reputation. The so-called neo-Objectivism of David Kelley and kindred spirits will only serve to sanitize Rand and make her works dull. Her philosophy may be a bit mad, but that is the only thing that provides any interest. A sanitized Rand is a boring Rand.